{"id":27,"date":"2021-02-03T20:17:54","date_gmt":"2021-02-03T20:17:54","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/exercises-for-1-1\/"},"modified":"2021-04-08T02:22:19","modified_gmt":"2021-04-08T02:22:19","slug":"module-1-review-problems","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/module-1-review-problems\/","title":{"raw":"Module 1 Review Problems","rendered":"Module 1 Review Problems"},"content":{"raw":"

True or False<\/em>? Justify your answers with a proof or a counterexample (1-4).<\/p>\r\n\r\n

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1.\u00a0<\/strong>A function is always one-to-one.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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2.\u00a0<\/strong>[latex]f \\circ g=g\\circ f[\/latex], assuming [latex]f[\/latex] and [latex]g[\/latex] are functions.<\/p>\r\n[reveal-answer q=\"fs-id1170572213137\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572213137\"]\r\n

False<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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3.\u00a0<\/strong>A relation that passes the horizontal and vertical line tests is a one-to-one function.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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4.\u00a0<\/strong>A relation passing the horizontal line test is a function.<\/p>\r\n[reveal-answer q=\"fs-id1170572237266\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572237266\"]\r\n

False<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following problems (5-8), state the domain and range of the given functions:<\/p>\r\n

[latex]f=x^2+2x-3,\\phantom{\\rule{3em}{0ex}}g=\\ln(x-5),\\phantom{\\rule{3em}{0ex}}h=\\dfrac{1}{x+4}[\/latex]<\/p>\r\n\r\n

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5.\u00a0<\/strong>[latex]h[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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6.\u00a0<\/strong>[latex]g[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572237397\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572237397\"]\r\n

Domain: [latex]x>5[\/latex], Range: all real numbers<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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7.\u00a0<\/strong>[latex]h\\circ f[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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8.\u00a0<\/strong>[latex]g\\circ f[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572237479\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572237479\"]\r\n

Domain: [latex]x>2[\/latex] or [latex]x<-4[\/latex], Range: all real numbers<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

Find the degree, [latex]y[\/latex]-intercept, and zeros for the following polynomial functions (9-10).<\/p>\r\n\r\n

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9.\u00a0<\/strong>[latex]f(x)=2x^2+9x-5[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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10.\u00a0<\/strong>[latex]f(x)=x^3+2x^2-2x[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572212690\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572212690\"]\r\n

Degree of 3, [latex]y[\/latex]-intercept: 0, Zeros: 0, [latex]\\sqrt{3}-1, \\, -1-\\sqrt{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

Simplify the following trigonometric expressions (11-12).<\/p>\r\n\r\n

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11.\u00a0<\/strong>[latex]\\dfrac{\\tan^2 x}{\\sec^2 x}+\\cos^2 x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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12.\u00a0<\/strong>[latex] \\cos^{2} x=\\sin^2 x[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572212808\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572212808\"]\r\n

[latex] \\cos(2x)[\/latex]\u00a0 or\u00a0 [latex]\\frac{1-2\\sin^2 x}{2}(2x)[\/latex]\u00a0 or\u00a0 [latex]\\frac{2\\cos^2 x -1}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

Solve the following trigonometric equations (13-14) on the interval [latex]\\theta =[-2\\pi ,2\\pi][\/latex] exactly.<\/p>\r\n\r\n

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13.\u00a0<\/strong>[latex]6\\cos^2 x-3=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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14.\u00a0<\/strong>[latex]\\sec^2 x-2\\sec x+1=0[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572243293\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572243293\"]\r\n

[latex]0, \\, \\pm 2\\pi [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

Solve the following logarithmic equations (15-16).<\/p>\r\n\r\n

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15.\u00a0<\/strong>[latex]5^x=16[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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16.\u00a0<\/strong>[latex]\\log_2 (x+4)=3[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572243408\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572243408\"]\r\n

4<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

Are the following functions one-to-one over their domain of existence? Does the function have an inverse? If so, find the inverse [latex]f^{-1}(x)[\/latex] of the function. Justify your answers (17-18).<\/p>\r\n\r\n

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17.\u00a0<\/strong>[latex]f(x)=x^2+2x+1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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18.\u00a0<\/strong>[latex]f(x)=\\dfrac{1}{x}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572139148\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572139148\"]\r\n

One-to-one; yes, the function has an inverse; inverse: [latex]f^{-1}(x)=\\frac{1}{x}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following problems (19-20), determine the largest domain on which the function is one-to-one and find the inverse on that domain.<\/p>\r\n\r\n

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19.\u00a0<\/strong>[latex]f(x)=\\sqrt{9-x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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20.\u00a0<\/strong>[latex]f(x)=x^2+3x+4[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572139305\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572139305\"]\r\n

[latex]x \\ge -\\frac{3}{2}, \\, f^{-1}(x)=-\\frac{3}{2}+\\frac{1}{2}\\sqrt{4y-7}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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21.\u00a0<\/strong>A car is racing along a circular track with diameter of 1 mi. A trainer standing in the center of the circle marks his progress every 5 sec. After 5 sec, the trainer has to turn [latex]55^{\\circ}[\/latex] to keep up with the car. How fast is the car traveling?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following problems (22-23), consider a restaurant owner who wants to sell T-shirts advertising his brand. He recalls that there is a fixed cost and variable cost, although he does not remember the values. He does know that the T-shirt printing company charges $440 for 20 shirts and $1000 for 100 shirts.<\/p>\r\n\r\n

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22. (a)<\/strong>\u00a0Find the equation [latex]C=f(x)[\/latex] that describes the total cost as a function of number of shirts and (b)<\/strong> determine how many shirts he must sell to break even if he sells the shirts for $10 each.<\/p>\r\n[reveal-answer q=\"fs-id1170572211085\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572211085\"]\r\n

a. [latex]C(x)=300+7x[\/latex] b. 100 shirts<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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23. (a)<\/strong> Find the inverse function [latex]x=f^{-1}(C)[\/latex] and describe the meaning of this function. (b)<\/strong> Determine how many shirts the owner can buy if he has $8000 to spend.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following problems (24-25), consider the population of Ocean City, New Jersey, which is cyclical by season.<\/p>\r\n\r\n

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24.\u00a0<\/strong>The population can be modeled by [latex]P(t)=82.5-67.5\\cos \\left[\\left(\\frac{\\pi}{6}\\right)t\\right][\/latex], where [latex]t[\/latex] is time in months ([latex]t=0[\/latex] represents January 1) and [latex]P[\/latex] is population (in thousands). During a year, in what intervals is the population less than 20,000? During what intervals is the population more than 140,000?<\/p>\r\n[reveal-answer q=\"fs-id1170572211270\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572211270\"]\r\n

The population is less than 20,000 from December 8 through January 23 and more than 140,000 from May 29 through August 2<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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25.\u00a0<\/strong>In reality, the overall population is most likely increasing or decreasing throughout each year. Let\u2019s reformulate the model as [latex]P(t)=82.5-67.5\\cos \\left[\\left(\\frac{\\pi}{6}\\right)t\\right]+t[\/latex], where [latex]t[\/latex] is time in months ([latex]t=0[\/latex] represents January 1) and [latex]P[\/latex] is population (in thousands). When is the first time the population reaches 200,000?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following problems (26-27), consider radioactive dating. A human skeleton is found in an archeological dig. Carbon dating is implemented to determine how old the skeleton is by using the equation [latex]y=e^{rt}[\/latex], where [latex]y[\/latex] is the percentage of radiocarbon still present in the material, [latex]t[\/latex] is the number of years passed, and [latex]r=-0.0001210[\/latex] is the decay rate of radiocarbon.<\/p>\r\n\r\n

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26.\u00a0<\/strong>If the skeleton is expected to be 2000 years old, what percentage of radiocarbon should be present?<\/p>\r\n[reveal-answer q=\"fs-id1170572544534\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572544534\"]\r\n

78.51%<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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27.\u00a0<\/strong>Find the inverse of the carbon-dating equation. What does it mean? If there is 25% radiocarbon, how old is the skeleton?<\/p>\r\n\r\n<\/div>\r\n<\/div>","rendered":"

True or False<\/em>? Justify your answers with a proof or a counterexample (1-4).<\/p>\n

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1.\u00a0<\/strong>A function is always one-to-one.<\/p>\n<\/div>\n<\/div>\n

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2.\u00a0<\/strong>[latex]f \\circ g=g\\circ f[\/latex], assuming [latex]f[\/latex] and [latex]g[\/latex] are functions.<\/p>\n

Show Solution<\/span><\/p>\n
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False<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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3.\u00a0<\/strong>A relation that passes the horizontal and vertical line tests is a one-to-one function.<\/p>\n<\/div>\n<\/div>\n

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4.\u00a0<\/strong>A relation passing the horizontal line test is a function.<\/p>\n

Show Solution<\/span><\/p>\n
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False<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following problems (5-8), state the domain and range of the given functions:<\/p>\n

[latex]f=x^2+2x-3,\\phantom{\\rule{3em}{0ex}}g=\\ln(x-5),\\phantom{\\rule{3em}{0ex}}h=\\dfrac{1}{x+4}[\/latex]<\/p>\n

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5.\u00a0<\/strong>[latex]h[\/latex]<\/p>\n<\/div>\n<\/div>\n

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6.\u00a0<\/strong>[latex]g[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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Domain: [latex]x>5[\/latex], Range: all real numbers<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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7.\u00a0<\/strong>[latex]h\\circ f[\/latex]<\/p>\n<\/div>\n<\/div>\n

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8.\u00a0<\/strong>[latex]g\\circ f[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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Domain: [latex]x>2[\/latex] or [latex]x<-4[\/latex], Range: all real numbers<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

Find the degree, [latex]y[\/latex]-intercept, and zeros for the following polynomial functions (9-10).<\/p>\n

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9.\u00a0<\/strong>[latex]f(x)=2x^2+9x-5[\/latex]<\/p>\n<\/div>\n<\/div>\n

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10.\u00a0<\/strong>[latex]f(x)=x^3+2x^2-2x[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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Degree of 3, [latex]y[\/latex]-intercept: 0, Zeros: 0, [latex]\\sqrt{3}-1, \\, -1-\\sqrt{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

Simplify the following trigonometric expressions (11-12).<\/p>\n

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11.\u00a0<\/strong>[latex]\\dfrac{\\tan^2 x}{\\sec^2 x}+\\cos^2 x[\/latex]<\/p>\n<\/div>\n<\/div>\n

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12.\u00a0<\/strong>[latex]\\cos^{2} x=\\sin^2 x[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]\\cos(2x)[\/latex]\u00a0 or\u00a0 [latex]\\frac{1-2\\sin^2 x}{2}(2x)[\/latex]\u00a0 or\u00a0 [latex]\\frac{2\\cos^2 x -1}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

Solve the following trigonometric equations (13-14) on the interval [latex]\\theta =[-2\\pi ,2\\pi][\/latex] exactly.<\/p>\n

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13.\u00a0<\/strong>[latex]6\\cos^2 x-3=0[\/latex]<\/p>\n<\/div>\n<\/div>\n

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14.\u00a0<\/strong>[latex]\\sec^2 x-2\\sec x+1=0[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]0, \\, \\pm 2\\pi[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

Solve the following logarithmic equations (15-16).<\/p>\n

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15.\u00a0<\/strong>[latex]5^x=16[\/latex]<\/p>\n<\/div>\n<\/div>\n

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16.\u00a0<\/strong>[latex]\\log_2 (x+4)=3[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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4<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

Are the following functions one-to-one over their domain of existence? Does the function have an inverse? If so, find the inverse [latex]f^{-1}(x)[\/latex] of the function. Justify your answers (17-18).<\/p>\n

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17.\u00a0<\/strong>[latex]f(x)=x^2+2x+1[\/latex]<\/p>\n<\/div>\n<\/div>\n

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18.\u00a0<\/strong>[latex]f(x)=\\dfrac{1}{x}[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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One-to-one; yes, the function has an inverse; inverse: [latex]f^{-1}(x)=\\frac{1}{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following problems (19-20), determine the largest domain on which the function is one-to-one and find the inverse on that domain.<\/p>\n

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19.\u00a0<\/strong>[latex]f(x)=\\sqrt{9-x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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20.\u00a0<\/strong>[latex]f(x)=x^2+3x+4[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]x \\ge -\\frac{3}{2}, \\, f^{-1}(x)=-\\frac{3}{2}+\\frac{1}{2}\\sqrt{4y-7}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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21.\u00a0<\/strong>A car is racing along a circular track with diameter of 1 mi. A trainer standing in the center of the circle marks his progress every 5 sec. After 5 sec, the trainer has to turn [latex]55^{\\circ}[\/latex] to keep up with the car. How fast is the car traveling?<\/p>\n<\/div>\n<\/div>\n

For the following problems (22-23), consider a restaurant owner who wants to sell T-shirts advertising his brand. He recalls that there is a fixed cost and variable cost, although he does not remember the values. He does know that the T-shirt printing company charges $440 for 20 shirts and $1000 for 100 shirts.<\/p>\n

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22. (a)<\/strong>\u00a0Find the equation [latex]C=f(x)[\/latex] that describes the total cost as a function of number of shirts and (b)<\/strong> determine how many shirts he must sell to break even if he sells the shirts for $10 each.<\/p>\n

Show Solution<\/span><\/p>\n
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a. [latex]C(x)=300+7x[\/latex] b. 100 shirts<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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23. (a)<\/strong> Find the inverse function [latex]x=f^{-1}(C)[\/latex] and describe the meaning of this function. (b)<\/strong> Determine how many shirts the owner can buy if he has $8000 to spend.<\/p>\n<\/div>\n<\/div>\n

For the following problems (24-25), consider the population of Ocean City, New Jersey, which is cyclical by season.<\/p>\n

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24.\u00a0<\/strong>The population can be modeled by [latex]P(t)=82.5-67.5\\cos \\left[\\left(\\frac{\\pi}{6}\\right)t\\right][\/latex], where [latex]t[\/latex] is time in months ([latex]t=0[\/latex] represents January 1) and [latex]P[\/latex] is population (in thousands). During a year, in what intervals is the population less than 20,000? During what intervals is the population more than 140,000?<\/p>\n

Show Solution<\/span><\/p>\n
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The population is less than 20,000 from December 8 through January 23 and more than 140,000 from May 29 through August 2<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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25.\u00a0<\/strong>In reality, the overall population is most likely increasing or decreasing throughout each year. Let\u2019s reformulate the model as [latex]P(t)=82.5-67.5\\cos \\left[\\left(\\frac{\\pi}{6}\\right)t\\right]+t[\/latex], where [latex]t[\/latex] is time in months ([latex]t=0[\/latex] represents January 1) and [latex]P[\/latex] is population (in thousands). When is the first time the population reaches 200,000?<\/p>\n<\/div>\n<\/div>\n

For the following problems (26-27), consider radioactive dating. A human skeleton is found in an archeological dig. Carbon dating is implemented to determine how old the skeleton is by using the equation [latex]y=e^{rt}[\/latex], where [latex]y[\/latex] is the percentage of radiocarbon still present in the material, [latex]t[\/latex] is the number of years passed, and [latex]r=-0.0001210[\/latex] is the decay rate of radiocarbon.<\/p>\n

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26.\u00a0<\/strong>If the skeleton is expected to be 2000 years old, what percentage of radiocarbon should be present?<\/p>\n

Show Solution<\/span><\/p>\n
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78.51%<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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27.\u00a0<\/strong>Find the inverse of the carbon-dating equation. What does it mean? If there is 25% radiocarbon, how old is the skeleton?<\/p>\n<\/div>\n<\/div>\n\n\t\t\t

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Candela Citations<\/h3>\n\t\t\t\t\t
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CC licensed content, Shared previously<\/div>